The Agencies Method for Coalition
Formation in Experimental Games
John Nash (University of Princeton)
Rosemarie Nagel (Universitat Pompeu Fabra, ICREA, BGSE)
Axel Ockenfels (University of Cologne)
Reinhard Selten (University of Bonn)
Stony Brook 2013
Motivation
• How to reach cooperation in a world of unequal bargaining
circumstances (based on Nash JF (2008) The agencies method for
modeling coalitions and cooperation in games Int Game Theory Rev 10(4):539–564)
– Repeated interaction and acceptance of agencies through a
voting mechanism
• Combination of non-cooperative and cooperative game theory
– Coalition formation, selection of agencies through
non-cooperative rules
– Multiplicity of non-cooperative equilibria
• A way out of multiplicity: structuring the strategy space through
cooperative solution concepts (e.g. Shapley value, nucleolous) and
equal split
• Run experiments letting behavior determine the outcome
Experimental bargaining procedure
• In a two step procedure an active player
decides whether or not to accept another
player as his agent. The final agent divides
the coalition value.
– If there are ties between accepted agents
then a random draw decides who becomes
the (final) agent.
– If nobody accepts another agent then the
procedure is repeated or a random stopping
rule terminates the round with zero payoffs or
two person coalition payoff
Bargaining Procedure
Start
Phase I
1
Every player accepts at
most one other player.
2
3
No
4
Yes
Is there an
eligible pair?
No
Stop?
Yes with prob.
1/100
Random selection of an
eligible pair (X,Y)
7
X and Z do or do not
accept the other active
player Z or X
8
Yes
No
10
No
Stop?
Yes with prob.
1/100
9
Is (X,Z) or
(Z,X) an
eligible pair?
Yes
Yes
Random selection of an
eligible pair (U,V) of X
and Z
Two person coalition
No coalition
Final payoffs zero:
pA = p B = p C = 0
5
X chooses final
payoff division
(pX, pY) of v(X,Y)
pZ =0
Phase II
13
Grand Coalition
11
U chooses final
payoff division
(pA,pB,pC) of v(ABC)
6
12
15
End
End
End
14
Phase III
Experimental design
• In every period an agency is voted
for (who divides the coalition value)
• The grand coalition always has
value 120.
• 3 subjects per group
• 10 independent groups per game
• 40 periods
• Maintain same player role
in same group and same game
• All periods are paid
Characteristic function games
games
1
2
3
4
5
6
7
8
9
10
v(12)
120
120
120
120
100
100
100
90
90
70
v(13)
100
100
100
100
90
90
90
70
70
50
v(23)
90
70
50
30
70
50
30
50
30
30
Game 1 - 5: no core
Theoretical solutions
• Non cooperative solutions
– One shot game: any coalition can be an
equilibrium outcome with any final agent
demanding coalition value for himself
In supergame any payoff division can be
equilibrium division
• Cooperative solution concepts
– We discuss Shapley value and Nucleolous
• Equal split as a good descriptive theory
Average results
and
cooperative solution concepts
Example GAME 6
Average group results (“+” = one group)
3
Equal split
Shapley value
90
50
++
Nucleolus
1
100
Game 6
2
Single group results
over time
Payoffs over time for all three players for each group, game 10
2
3
4
5
6
7
8
0
40
80
120
0
40
80
120
1
0
20
30
40
0
10
20
30
40
10
80
120
9
10
0
40
Game 10
V(1,2)
= 70
0
10
20
30
40
0
10
20
30
40
time
Payoffs 1
Graphs by Group
Payoffs 2
Payoffs 3
V(1,3)
= 50
V(2,3)
= 30
number of observations
(out of 100 groups)
35
30
25
Number of times of equal
split in each group, e.g.
30% of all groups divide
fairly in 36- 40 rounds
=> High heterogeneity
20
15
10
5
0
0-3
4-7
8-11 12-15 16-19 20-23 24-27 28-31 32-35 36-40
number of periods (out of 40) with equal split divisions
Average vector of Strong
player division in game 6
Average payoff vectors across all
3
periods in game 6
C
Many near equal split
90
1
50
100
Game 6
90
Many near Shapl.
Value, nucleolous
50
2
A
100
Game 6
B
Why is there equalization of payoffs over time,
given that the strong player demands on
average very much for himself within a single
period?
Equalization through reciprocity and balancing of
power through voting mechanism
Payoff offers between A&B or
A&C or B&C
What final agents offer to each other:
Rank correlation significantly positive:
If you offer “high” to me I offer “high” to
you and similar with “low” offers
=> Equalization across periods
THROUGH RECIPROCITY
Number of times being agent (out of 40)
and own payoff demand
If you demand too much for
yourself, less likely to be voted as
final agent
Equalization across periods
THROUGH balance of power
Conclusion
• A theoretical model to reach cooperation in three
person coalition formation using
– a non-cooperative model of interacting players
– implement experiments
• Both the Shapley value and the nucleolus (cooperative
concepts) seem to give comparatively more payoff
advantage to player 1 than would appear to be the
implication of the average results across periods
derived directly from the experiments.
• Equalization of payoffs through reciprocity and
balance of power.